Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $n = \dfrac{4x}{14x + 35} \div \dfrac{8x}{10x + 25} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{4x}{14x + 35} \times \dfrac{10x + 25}{8x} $ When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 4x \times (10x + 25) } { (14x + 35) \times 8x } $ $ n = \dfrac {4x \times 5(2x + 5)} {8x \times 7(2x + 5)} $ $ n = \dfrac{20x(2x + 5)}{56x(2x + 5)} $ We can cancel the $2x + 5$ so long as $2x + 5 \neq 0$ Therefore $x \neq -\dfrac{5}{2}$ $n = \dfrac{20x \cancel{(2x + 5})}{56x \cancel{(2x + 5)}} = \dfrac{20x}{56x} = \dfrac{5}{14} $